direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary
Aliases: C11×D16, C176⋊5C2, C16⋊1C22, D8⋊1C22, C22.15D8, C44.36D4, C88.24C22, (C11×D8)⋊5C2, C8.2(C2×C22), C4.1(D4×C11), C2.3(C11×D8), SmallGroup(352,60)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C11×D16
G = < a,b,c | a11=b16=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C11×D16
►On 176 points
Generators in S176
(1 65 122 26 52 149 176 133 89 103 48)(2 66 123 27 53 150 161 134 90 104 33)(3 67 124 28 54 151 162 135 91 105 34)(4 68 125 29 55 152 163 136 92 106 35)(5 69 126 30 56 153 164 137 93 107 36)(6 70 127 31 57 154 165 138 94 108 37)(7 71 128 32 58 155 166 139 95 109 38)(8 72 113 17 59 156 167 140 96 110 39)(9 73 114 18 60 157 168 141 81 111 40)(10 74 115 19 61 158 169 142 82 112 41)(11 75 116 20 62 159 170 143 83 97 42)(12 76 117 21 63 160 171 144 84 98 43)(13 77 118 22 64 145 172 129 85 99 44)(14 78 119 23 49 146 173 130 86 100 45)(15 79 120 24 50 147 174 131 87 101 46)(16 80 121 25 51 148 175 132 88 102 47)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128)(129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144)(145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)(161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176)
(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 18)(19 32)(20 31)(21 30)(22 29)(23 28)(24 27)(25 26)(33 46)(34 45)(35 44)(36 43)(37 42)(38 41)(39 40)(47 48)(49 54)(50 53)(51 52)(55 64)(56 63)(57 62)(58 61)(59 60)(65 80)(66 79)(67 78)(68 77)(69 76)(70 75)(71 74)(72 73)(81 96)(82 95)(83 94)(84 93)(85 92)(86 91)(87 90)(88 89)(97 108)(98 107)(99 106)(100 105)(101 104)(102 103)(109 112)(110 111)(113 114)(115 128)(116 127)(117 126)(118 125)(119 124)(120 123)(121 122)(129 136)(130 135)(131 134)(132 133)(137 144)(138 143)(139 142)(140 141)(145 152)(146 151)(147 150)(148 149)(153 160)(154 159)(155 158)(156 157)(161 174)(162 173)(163 172)(164 171)(165 170)(166 169)(167 168)(175 176)
G:=sub<Sym(176)| (1,65,122,26,52,149,176,133,89,103,48)(2,66,123,27,53,150,161,134,90,104,33)(3,67,124,28,54,151,162,135,91,105,34)(4,68,125,29,55,152,163,136,92,106,35)(5,69,126,30,56,153,164,137,93,107,36)(6,70,127,31,57,154,165,138,94,108,37)(7,71,128,32,58,155,166,139,95,109,38)(8,72,113,17,59,156,167,140,96,110,39)(9,73,114,18,60,157,168,141,81,111,40)(10,74,115,19,61,158,169,142,82,112,41)(11,75,116,20,62,159,170,143,83,97,42)(12,76,117,21,63,160,171,144,84,98,43)(13,77,118,22,64,145,172,129,85,99,44)(14,78,119,23,49,146,173,130,86,100,45)(15,79,120,24,50,147,174,131,87,101,46)(16,80,121,25,51,148,175,132,88,102,47), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,18)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)(47,48)(49,54)(50,53)(51,52)(55,64)(56,63)(57,62)(58,61)(59,60)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)(81,96)(82,95)(83,94)(84,93)(85,92)(86,91)(87,90)(88,89)(97,108)(98,107)(99,106)(100,105)(101,104)(102,103)(109,112)(110,111)(113,114)(115,128)(116,127)(117,126)(118,125)(119,124)(120,123)(121,122)(129,136)(130,135)(131,134)(132,133)(137,144)(138,143)(139,142)(140,141)(145,152)(146,151)(147,150)(148,149)(153,160)(154,159)(155,158)(156,157)(161,174)(162,173)(163,172)(164,171)(165,170)(166,169)(167,168)(175,176)>;
G:=Group( (1,65,122,26,52,149,176,133,89,103,48)(2,66,123,27,53,150,161,134,90,104,33)(3,67,124,28,54,151,162,135,91,105,34)(4,68,125,29,55,152,163,136,92,106,35)(5,69,126,30,56,153,164,137,93,107,36)(6,70,127,31,57,154,165,138,94,108,37)(7,71,128,32,58,155,166,139,95,109,38)(8,72,113,17,59,156,167,140,96,110,39)(9,73,114,18,60,157,168,141,81,111,40)(10,74,115,19,61,158,169,142,82,112,41)(11,75,116,20,62,159,170,143,83,97,42)(12,76,117,21,63,160,171,144,84,98,43)(13,77,118,22,64,145,172,129,85,99,44)(14,78,119,23,49,146,173,130,86,100,45)(15,79,120,24,50,147,174,131,87,101,46)(16,80,121,25,51,148,175,132,88,102,47), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,18)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)(47,48)(49,54)(50,53)(51,52)(55,64)(56,63)(57,62)(58,61)(59,60)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)(81,96)(82,95)(83,94)(84,93)(85,92)(86,91)(87,90)(88,89)(97,108)(98,107)(99,106)(100,105)(101,104)(102,103)(109,112)(110,111)(113,114)(115,128)(116,127)(117,126)(118,125)(119,124)(120,123)(121,122)(129,136)(130,135)(131,134)(132,133)(137,144)(138,143)(139,142)(140,141)(145,152)(146,151)(147,150)(148,149)(153,160)(154,159)(155,158)(156,157)(161,174)(162,173)(163,172)(164,171)(165,170)(166,169)(167,168)(175,176) );
G=PermutationGroup([[(1,65,122,26,52,149,176,133,89,103,48),(2,66,123,27,53,150,161,134,90,104,33),(3,67,124,28,54,151,162,135,91,105,34),(4,68,125,29,55,152,163,136,92,106,35),(5,69,126,30,56,153,164,137,93,107,36),(6,70,127,31,57,154,165,138,94,108,37),(7,71,128,32,58,155,166,139,95,109,38),(8,72,113,17,59,156,167,140,96,110,39),(9,73,114,18,60,157,168,141,81,111,40),(10,74,115,19,61,158,169,142,82,112,41),(11,75,116,20,62,159,170,143,83,97,42),(12,76,117,21,63,160,171,144,84,98,43),(13,77,118,22,64,145,172,129,85,99,44),(14,78,119,23,49,146,173,130,86,100,45),(15,79,120,24,50,147,174,131,87,101,46),(16,80,121,25,51,148,175,132,88,102,47)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128),(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144),(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160),(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176)], [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9),(17,18),(19,32),(20,31),(21,30),(22,29),(23,28),(24,27),(25,26),(33,46),(34,45),(35,44),(36,43),(37,42),(38,41),(39,40),(47,48),(49,54),(50,53),(51,52),(55,64),(56,63),(57,62),(58,61),(59,60),(65,80),(66,79),(67,78),(68,77),(69,76),(70,75),(71,74),(72,73),(81,96),(82,95),(83,94),(84,93),(85,92),(86,91),(87,90),(88,89),(97,108),(98,107),(99,106),(100,105),(101,104),(102,103),(109,112),(110,111),(113,114),(115,128),(116,127),(117,126),(118,125),(119,124),(120,123),(121,122),(129,136),(130,135),(131,134),(132,133),(137,144),(138,143),(139,142),(140,141),(145,152),(146,151),(147,150),(148,149),(153,160),(154,159),(155,158),(156,157),(161,174),(162,173),(163,172),(164,171),(165,170),(166,169),(167,168),(175,176)]])
121 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 8A | 8B | 11A | ··· | 11J | 16A | 16B | 16C | 16D | 22A | ··· | 22J | 22K | ··· | 22AD | 44A | ··· | 44J | 88A | ··· | 88T | 176A | ··· | 176AN |
| order | 1 | 2 | 2 | 2 | 4 | 8 | 8 | 11 | ··· | 11 | 16 | 16 | 16 | 16 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 | 88 | ··· | 88 | 176 | ··· | 176 |
| size | 1 | 1 | 8 | 8 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 8 | ··· | 8 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
121 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C11 | C22 | C22 | D4 | D8 | D16 | D4×C11 | C11×D8 | C11×D16 |
| kernel | C11×D16 | C176 | C11×D8 | D16 | C16 | D8 | C44 | C22 | C11 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 10 | 10 | 20 | 1 | 2 | 4 | 10 | 20 | 40 |
Matrix representation of C11×D16 ►in GL2(𝔽353) generated by
| 131 | 0 |
| 0 | 131 |
| 150 | 340 |
| 183 | 163 |
| 150 | 340 |
| 20 | 203 |
G:=sub<GL(2,GF(353))| [131,0,0,131],[150,183,340,163],[150,20,340,203] >;
C11×D16 in GAP, Magma, Sage, TeX
C_{11}\times D_{16} % in TeX
G:=Group("C11xD16"); // GroupNames label
G:=SmallGroup(352,60);
// by ID
G=gap.SmallGroup(352,60);
# by ID
G:=PCGroup([6,-2,-2,-11,-2,-2,-2,553,3171,1593,165,7924,3970,88]);
// Polycyclic
G:=Group<a,b,c|a^11=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export